Textbook Question
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.88a
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
Verified step by step guidance1
Identify the region described in Exercise 87, including the functions and the interval over which the region is defined. This is essential because the volume depends on the shape of the region being revolved.
Set up the volume integral using the disk or washer method since the solid is generated by revolving the region about the x-axis. The general formula for the volume is \(V = \pi \int_a^b [R(x)]^2 \, dx\), where \(R(x)\) is the radius of the disk at position \(x\).
Determine the radius function \(R(x)\), which is the distance from the x-axis to the curve defining the boundary of the region. If there are two curves, use the difference of their squares to form the washer method: \(V = \pi \int_a^b \left([R_{outer}(x)]^2 - [R_{inner}(x)]^2\right) \, dx\).
Substitute the expressions for \(R(x)\) (or \(R_{outer}(x)\) and \(R_{inner}(x)\)) and the limits of integration \(a\) and \(b\) into the integral to form the definite integral representing the volume.
Evaluate the integral (or set it up for evaluation) to find the volume of the solid. This may involve integrating polynomial, trigonometric, or other functions depending on the original region.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume is typically calculated using integral methods such as the disk or washer method, which sum up infinitesimal cross-sectional areas along the axis of rotation.
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04:48
Finding Volume Using Disks
Disk and Washer Methods
The disk method calculates volume by integrating the area of circular cross-sections perpendicular to the axis of rotation. The washer method extends this by subtracting inner radii when the solid has a hollow part, useful when the region is bounded by two curves.
Recommended video:
06:30Disk Method Using y-Axis
Setting up Definite Integrals
To find the volume, one must correctly set up the definite integral with proper limits and integrand representing the radius (or radii) of the cross-sections. This requires understanding the region's boundaries and expressing the radius as a function of the variable of integration.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of sin(x + 1) dx
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
b. π ≤ x ≤ 2π.
1
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Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 1) dx